<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>For an arbitrary <span class="process-math">\({\bf A}\text{,}\)</span> we shall try to use transforms to obtain a diagonal matrix. We first need to find a linear independent eigenvectors <span class="process-math">\(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \cdots, \vec{\xi}^{(n)}\)</span> (corresponding to <span class="process-math">\(r_1, r_2, \cdots, r_n\)</span>). Denote</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf T}=(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \cdots, \vec{\xi}^{(n)})=\left(
\begin{array}{cccc}
\xi^{(1)}_1 &amp; \xi^{(2)}_1 &amp; \cdots &amp; \xi^{(n)}_1 \\
\xi^{(1)}_2 &amp; \xi^{(2)}_2 &amp; \cdots &amp; \xi^{(n)}_2 \\
\vdots &amp; \vdots &amp;   &amp; \vdots \\
\xi^{(1)}_n &amp; \xi^{(2)}_n &amp; \cdots &amp; \xi^{(n)}_n \\
\end{array}
\right).
\end{equation*}
</div>
<span class="incontext"><a href="sec6_5.html#p-281" class="internal">in-context</a></span>
</body>
</html>
